Yes. Healthcare organizations are permitted to send SMS text messages to patients as long as:

1. The patient has given express consent.
2. The message contains healthcare-related content, specifically: appointment and exam confirmations and reminders, wellness checkups, hospital pre-registration instructions, pre-operative instructions, lab results, post-discharge follow-ups intended to prevent readmission, prescription notifications, and home healthcare.
3. The message does not include telemarketing, solicitation, or advertising content, or include accounting, billing, debt-collection, or other financial content.

Regarding express consent, on July 10, 2015, the FCC issued a Declaratory Ruling and Order of the Telephone Consumer Protection Act (TCPA) that clarified express consent:

We clarify, therefore, that the provision of a phone number to a healthcare provider constitutes prior express consent for healthcare calls subject to HIPAA by a HIPAA-covered entity and business associates acting on its behalf, as defined by HIPAA, if the covered entities and business associates are making calls within the scope of the consent given, and absent instructions to the contrary.

In other words, the act of a patient giving his / her phone number to a healthcare organization counts as express consent to be called. Note that the TCPA treats SMS texts and voice calls as the same.